It looks like the following quantity $$ q(k)=\frac{k+1}{2k}(1+\log k) - \sum_{i=2}^k \frac{i}{k^2} \log i $$ tends to $3/4$ as $k$ goes to infinity.
Is there a nice way to prove it?
2026-05-15 13:45:40.1778852740
on the convergence of an infinite series involving logarithms
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We have $$ \sum_{i=1}^k \frac{i}{k^2} \log i=\frac1k\sum_{i=1}^k \frac{i}{k}\log(i)=\frac1k\sum_{i=1}^k \frac{i}{k}(\log(\frac ik)+\log(k))=\frac1k\sum_{i=1}^k \frac{i}{k}\log(\frac ik)+\log(k)\frac1k\sum_{i=1}^k \frac{i}{k} $$ Now use Riemann integration definition.